#
Physics From Strings ^{1}^{1}1Invited talk
presented at PASCOS-98, Northeastern University, March 1998,
based on work done in collaboration
with J. Cleaver, M. Cvetič, D. Demir,
J. R. Espinosa, L. Everett, and J. Wang.

UPR-0806T, hep-ph/9805486

Many extensions of the standard model, especially grand unified theories and superstring models, predict the existence of additional bosons and associated exotic chiral supermultiplets. It has recently been argued that for classes of string motivated models with supergravity mediated supersymmetry breaking there are two scenarios for the additional s: either the mass is in the accessible range (1 TeV), providing a natural solution to the problem and implications for the Higgs and sparticle masses and for the LSP; or, when the breaking is associated with a -flat direction, at an intermediate scale, providing a possible explanation for the hierarchies of quark and charged lepton masses and new possibilities for neutrino masses. Related work, examining the detailed structure of specific perturbative string vacua for and -flat directions, surviving ’s and exotics, and effective couplings, is briefly described.

PAUL LANGACKER

PHYSICS FROM STRINGS ^{1}^{1}1Work in collaboration
with J. Cleaver, M. Cvetič, D. Demir,
J. R. Espinosa, L. Everett, and J. Wang.

Many extensions of the standard model, especially grand unified theories and superstring models, predict the existence of additional bosons and associated exotic chiral supermultiplets. It has recently been argued that for classes of string motivated models with supergravity mediated supersymmetry breaking there are two scenarios for the additional s: either the mass is in the accessible range (1 TeV), providing a natural solution to the problem and implications for the Higgs and sparticle masses and for the LSP; or, when the breaking is associated with a -flat direction, at an intermediate scale, providing a possible explanation for the hierarchies of quark and charged lepton masses and new possibilities for neutrino masses. Related work, examining the detailed structure of specific perturbative string vacua for and -flat directions, surviving ’s and exotics, and effective couplings, is briefly described.

## 1 Phenomenology

If the standard model (SM) gauge group is extended by an additional , then the mass eigenstates will be mixtures of the SM and new with mixing angle . There are stringent limits on and from precision pole and neutral current experiments [1], because: (i) is shifted from the SM prediction by mixing; (ii) the couplings are changed by the mixing; (iii) exchange may be important in neutral current amplitudes. There are also Tevatron [2] limits on from the non-observation of decays into or . The limits are model dependent, depending not only on the chiral couplings to , and , but (in the case of the direct production limits) on the number of open decay channels into exotics, superpartners, etc. Typically, for properties motivated by grand unification (GUTs) one finds 600 – 1000 GeV and few . For one expects , where and are respectively the ordinary and new gauge couplings ( is the weak hypercharge coupling), and depends on the Higgs charges under the and their VEVs. The most stringent limits on , which occur in those specific models in which is fixed, actually come from . For models with suppressed couplings to ordinary fermions [3], such as leptophobic models, much smaller is allowed (e.g., 150 GeV; one could even have , where is the boson that is mainly the SM one), as is larger few .

It should be possible to extend the direct limits on with GUT-type couplings to around a TeV at the Tevatron. At the LHC (with 100 fb), one should be able to discover a via its leptonic decays up to around 4 TeV [4, 1], well above the range 1 TeV expected in superstring theories [5]. At an NLC (500 GeV, 50 fb) one has . Observations of cross sections, forward-backward and polarization asymmetries, etc., should allow a sensitivity to a (virtual) up to 1-3 TeV, increasing rapidly with energy [4]. Once a is observed, one will want to determine not only its mass and mixing, but its chiral couplings to identify its origin. At the LHC, a combination of forward-backward asymmetries (as a function of rapidity), rapidity distributions, rare decays (), and associated production of should provide significant diagnostic ability up to 1 - 2 TeV [4], with the information provided by the LHC and NLC complementary.

## 2 String Motivated Models

It is well known that that electroweak (EW) breaking in the MSSM with supergravity mediated supersymmetry breaking (SUGRA) can be radiative; i.e., a positive Higgs mass square from SUSY breaking at the Planck scale can be driven negative at low energy due to the large Yukawa coupling associated with the quark. In perturbative string models there are often extra non-anomalous ’s which are not broken at the string scale. These can be broken radiatively [5], either at the electroweak scale (i.e., less than 1 TeV) [6], or, when the breaking is associated with a -flat direction, at an intermediate scale [7].

### 2.1 Electroweak Breaking

In the ordinary MSSM the potential for the two Higgs doublets is , where

(1) |

(A term in involving charged fields has been omitted.) The term is derived from the superpotential

(2) |

Unlike the ordinary SM, in which the quartic coefficient in the Higgs potential is arbitrary, the coefficient of the quartic term is associated with gauge couplings, leading to the the upper bound on the lightest Higgs scalar (tree level) or 130 GeV (including loops). The EW scale is , where , and . The scale of is set not only by the soft SUSY breaking parameters and in , but also by the supersymmetry preserving parameter .

In SUGRA models one assumes that SUSY is broken in a hidden sector at some intermediate scale and then transmitted to the observable sector by supergravity. The soft breaking parameters are then all of the same order of magnitude 1 TeV (e.g., if , where is the Planck scale, then GeV). In particular, all scalars in the theory (Higgs, squarks, sleptons) typically acquire positive mass squares of at . (Universal soft breaking, which we do not assume, is the stronger assumption that the scalar mass squares are all equal at .) This is the wrong sign for EW breaking. However, for sufficiently large the Yukawa interactions can drive negative (and of ) at low energies. Hence, radiative breaking requires a large , consistent with the experimental value 175 GeV.

Thus, the SUGRA mechanism can yield the needed soft parameters. However, one also requires . Since is a supersymmetric parameter, this requires fine-tuning in the context of the MSSM, the famous problem [8]. If one has some mechanism to force , then one can generate an effective by several mechanisms, including: (1) The Giudice-Masiero mechanism [9], in which is transmitted to the observable sector by SUGRA along with the soft breaking terms. (2) The NMSSM [10], in which one introduces a SM singlet field , with superpotential terms , so that . However, the cubic term, needed to avoid an axion, allows a discrete symmetry and undesirable cosmological domain walls. (3) An extra gauge symmetry [11] broken by the VEV of a SM singlet with can force with . Unlike the NMSSM there is no domain wall problem.

When the SUGRA MSSM is considered in the context of a class of perturbative string models, one obtains in addition: (1) by string selection rules. (2) There are typically additional non-anomalous ’s as well as exotic chiral supermultiplets (which can play a role in radiative breaking). (3) The Yukawa couplings at the string scale are either zero or , as needed for radiative breaking.

### 2.2 Symmetry Breaking with an Extra

An additional non-anomalous gauge symmetry can be broken by the VEV of a SM singlet with nonzero charge . The addition of the and to the ordinary SM results in new arbitrary parameters in the scalar potential, so there is in general no prediction for the mass scale. However, things are much more constrained in the extension of the MSSM [12]. Let us assume that , where are the charges of , so that invariance forces . If one can have

(3) |

where the last term is an optional coupling of to new exotic multiplets . The analogue of (1) becomes

(4) |

where is the gauge coupling. Thus, if acquires a VEV, one has an effective parameter , and the corresponding . Acceptable EW breaking can occur if and are of (TeV). If all of the soft SUSY breaking parameters are of (), then one expects not only and but also and to be of (). Only some limiting (or somewhat tuned) cases will yield allowed and . The mixing angle is given by

(5) |

where

(6) | |||||

(7) | |||||

(8) |

are respectively the and mass squares in the absence of mixing and the mixing mass squared, and . Small mixing requires small and/or .

Two viable scenarios were described in [6]. (i) In
the Large Scenario the EW and breaking is
driven by a large in the last term in (4).
This leads to (a generalization of ).
In the special case one finds
and the prediction .
For example, a concrete model [15] with the couplings of the
model ^{2}^{2}2This model, which has the matter content and couplings of
three 27-plets as well as two Higgs-like doublets from , is anomaly
free and consistent with gauge unification. It is string-motivated, i.e.,
the Yukawa couplings are of or zero, and the and violating
GUT Yukawa relations are not respected, so that can be light.
yields GeV. This model is not leptophobic ^{3}^{3}3An
alternative model involving matter from an extra 78 has much larger kinetic
mixing and can lead to leptophobic couplings [3].
and is therefore
excluded, but it illustrates a scenario that may be viable in string-derived
models with suppressed couplings to ordinary fermions.

(ii) In the Large Scenario one assumes that all of the soft parameters () are of (1 TeV), with . Then and . One can have a smaller EW scale by accidental cancellations (because of this often only involves one constraint). To avoid excessive tuning this implies (1 TeV). Then is small due to , and can be further suppressed for small .

Both scenarios have a number of interesting consequences. These include: (i) A solution to the problem [11], with naturally of (large ) or (TeV) (large ). (ii) A and associated exotics with masses (TeV). (iii) a predictive pattern of Higgs masses (large ) or weakened upper limit on the lightest Higgs (large ) [12]. (iv) Characteristic shifts in the scalar masses due to the term [17]. (v) New dark matter possibilities (e.g., ) [18].

The weak scale parameters needed for both scenarios can be generated by radiative breaking [6, 14, 15]. As motivated by SUGRA, we assume that at all of the scalar mass squares () are positive and of (), but not necessarily universal. We also assume that the gaugino masses and the parameters are of (). The coupled one-loop RGE equations for the running gauge and Yukawa couplings and the soft parameters were studied for various toy models [6] and models with couplings [15] to relate the initial parameters at to the EW scale parameters. It was found that the large scenario was possible though somewhat fine-tuned (it is necessary to ensure moderate , the term associated with , to avoid dangerous charge-color breaking minima). The large scenario, which requires at the EW scale is most easily obtained if there is a Yukawa coupling of to exotic multiplets (the optional term in (3)), but can be obtained without such couplings for some (non-universal) initial conditions.

### 2.3 Intermediate Scale Breaking

Another possibility is breaking associated with a
-flat direction at an intermediate scale [7],
which is expected to occur in many string models and which
may be associated with fermion mass hierarchies^{4}^{4}4A
similar mechanism could occur for a total gauge singlet field..
This can occur, for example, if there are two SM singlets
with . If the model
is also -flat at the renormalizable level (i.e., there
are no terms or in ), the low energy potential for is

(9) |

where the quartic term vanishes for .

As an example, suppose , and further that at low energies and , as would typically occur by the radiative mechanism if contains a term . If the minimum will occur at . Then, and will be at the EW scale ( 1 TeV), just as in the case of a single . On the other hand, for , the potential along the and flat direction is

(10) |

which appears to be unbounded from below. However, can be stabilized by either of two mechanisms [7]: (a) The leading loop corrections to the effective (RGE-improved) potential result in in (10). Since runs from a positive value at to a negative value at low energies, the RGE-improved potential will have a minimum close to but slightly below the scale at which goes through zero. It was shown in [7] that can occur anywhere in the range GeV, depending on the soft breaking parameters and the exotic () quantum numbers. (b) Another possibility is that the -flatness is lifted by higher-dimensional nonrenormalizable operators (NRO) in , as are expected in string models, such as , where GeV is of the order of the string scale. For example, if contains

(11) |

when evaluated along the flat direction , then the potential will be minimized at the scale

(12) |

which is around GeV for .

In general, both the radiative and NRO stabilization mechanisms can occur, and will be of the order of the smaller of and . In both cases, one expects . Also, is of at the minimum, leading to an electroweak scale invisible scalar. There are also characteristic -induced shifts in the effective soft masses [7]. A effective parameter can be generated by the superpotential term

(13) |

(The special case is needed for the EW scale breaking scenario.) For radiative stabilization, one obtains the needed 1 TeV for, e.g., and GeV. For NRO stabilization,

(14) |

which is of the order of the soft breaking (and EW) scale for .

Intermediate scale breaking scenarios have interesting implications for quark, charged lepton, and neutrino masses [7, 19]. For example, a -type quark mass may be generated by the term

(15) |

where in string models the nonzero coefficients are of for , and can be absorbed into for . (15) leads to an effective Yukawa coupling and fermion mass (in the case of NRO stabilization)

(16) |

Presumably, the mass is associated with [20], while the
and masses, and any inter-generational masses associated with
family mixing, could be due to operators of higher dimension. Similar
hierarchies of dimensions of operators could lead to small and
type masses and mixings, especially for the first two families, as
well as naturally tiny Dirac neutrino masses, without the need
for invoking a seesaw [7]. Which terms actually have
non-zero coefficients is determined not only by gauge invariance in the
four dimensional effective field theory, but by string selection rules
as well [21]. This mechanism of small effective Yukawas
suppressed by intermediate scale VEV’s is somewhat analogous to
the attempts [22, 23] to generate Yukawas suppressed by powers
of , where is a field which
breaks the anomalous present in many free fermionic models [24].
However, the lower intermediate scale considered here allows for the use of lower
dimension operators^{5}^{5}5Most of the studies [23] have assumed that
the non-zero coefficients could be classified according to the anomalous
symmetry. However, this is not the case in free fermionic
models [21].. It is also possible to generate Majorana masses
for sterile (-singlet) neutrinos by the operators

(17) |

implying

(18) |

which can be large (leading to a seesaw) or small, depending on the sign of . From (16) and (18) and the fact that for radiative breaking, one finds that neutrino Dirac and Majorana masses can be naturally small and comparable in the special case , where is the power analogous to in (15) for a Dirac neutrino mass term [19]. This can lead to significant mixing between ordinary neutrinos and light sterile neutrinos, as is suggested phenomenologically by the experimental hints of neutrino mass [25].

## 3 Perturbative String Vacua

The work discussed in Section 2 was motivated by certain general features of perturbative string models, especially (a) the existence of additional ’s and exotics, (b) that the Yukawa couplings at the string scale are either zero or of , (c) that world-sheet selection rules often forbid terms in the superpotential that would be allowed by the gauge symmetries of the effective four-dimensional field theory, and (d) that there are no elementary bilinear (mass) terms in . A more ambitious project is to try to derive the consequences of specific string vacua. There are many possible string vacua, and none that have been studied are fully realistic. However, there are models based on the free fermionic construction [26, 24] that are quasi-realistic, containing the ingredients of the MSSM (gauge group, and candidates for three ordinary families and two Higgs doublets) and some form of gauge unification. They typically also contain a (partially) hidden sector non-abelian group, an anomalous , a number of extra non-anomalous s, and many additional exotic chiral supermultiplets. The latter include non-chiral exotic multiplets, fractionally charged states, and mixed states transforming non-trivially under both the ordinary and hidden sector groups.

A first step in studying the low energy consequences of such models is to determine which fields acquire VEVs at or near the string scale, in a way that breaks the anomalous but maintains and flatness. Techniques have recently been developed to compute classes of -flat directions that can be proved -flat to all orders [21]. A number of models were considered, and it was found that those which have such flat directions composed on non-abelian singlet fields typically leave one or more non-anomalous s unbroken at the string scale. A next step, currently in progress [27], is to study the effective superpotential of the resulting model in these flat directions, after replacing the scalar fields which appear in the flat directions by their VEVs. In particular, it will be possible to study the breaking patterns and low energy consequences, after making appropiate ansätze for the soft supersymmery breaking terms and the structure of the Kähler potential. It is unlikely that any realistic models will be found, but it is hoped that the analysis will give useful insights into the type of physics consequences that may derive from perturbative string theories.

## Acknowledgments

It is a pleasure to thank J. Cleaver, M. Cvetič, D. Demir, J. R. Espinosa, L. Everett, and J. Wang for fruitful and enjoyable collaborations. This work was supported by U.S. Department of Energy Grant No. DOE-EY-76-02-3071.

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